My research began before we had a specific focus for the content of the lesson. Our group had discussed math agency and the relationship between language access and confidence. We reflected on mathematical discussion and the opportunities for students to share their thinking or engage with peer strategies. We knew that students that did not have the language to describe their strategies or will not feel confident to share or question at discourse time. We also knew that discussion structures can help all students access language and build peer engagement.  I began my investigation looking at ways to support motivation and build agency with whole class routines. My search led me to an article about social- psychological interventions (Yeager and Walton, 2011). This article was about efforts to try social- psychological interventions across classrooms and schools, scaling up personalized support to provide the benefits to all students. I was struck with the implications of small adjustments to instruction that can reinforce a growth mindset. I thought we could implement some small changes to the classroom discourse structure to provide resources or language that would help learners feel empowered to engage with peer thinking. 

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           The idea to scale up interventions was something I had seen first hand in math classrooms at High Tech Elementary schools. Teachers used a program called YouCubed to build mathematical mindsets with the whole class. With some multiple solution number talks and lessons about neuroplasticity, the launch of math story problems was a welcomed opportunity to take risks. I have also noticed mindset interventions usually live at the beginning of the school year. For our series of lessons we planned to make sure to highlight and track student conjectures and language. The opportunity for students to use their own words to describe their thinking and share this with the class would be a language scaffold and a way to help more students feel confident to build off of each other’s ideas.
At this time in our lesson study planning we were still on the fence about the structure of our math lesson. We had debated using the classes routine of cognitively guided instruction (CGI)  or a more open lesson structure like a number talk. Number talks are math discussions where students try to identify all the possible ways a problem can be solved. The focus is not on hte right answer but the connections between the possible solutions. 

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           CGI (Carpenter, 1999) is a student centered math practice that encourages students to build of existing schema and develop a deep number sense. CGI begins with a story problem that students read then retell. They are asked a carefully structured comprehension question and given brief opportunities to explain responses. Students then work independently or in a small group to solve the problem with as many strategies as they can come up with. The teacher will confer with students while they work to more deeply understand the thinking behind specific strategies and understandings. After selecting students to share a range of strategies, the teacher will gather students for a discussion. Students will take turns sharing their thinking and respond to questions,  comments, and connections. The teacher will chart the strategies and discussion highlights. The practice is usually organized by patterns in problem type so students can refer to their previous work in their reasoning.

         I was pushing for a number talk because I was excited about the idea of designing a lesson with a group of peers and the CGI problem types are currently sequenced. Our host teacher was inclined to stick with CGI because it is a staple in the students math instruction and something her class would benefit from. I was so excited to find an article on problem types and patterns of motivation (Daly et all., 2019). This article provided empirical evidence, using EEG monitors, that open ended problems encouraged motivation. This article helped me see the value in researching ways to build student thinking through their practiced routines.  

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          CGI is already open ended in the way students approach the story problem. I have seen a great range of strategies for a seemingly straightforward story problem. Our structure was motivating, but we still needed to find ways to scaffold discussion and get students thinking about peer strategies. Looking at student work along with anecdotes from our host teacher, we needed to help our focus students 2-4 find their voice and describe their thinking. I decided to ask experts. 

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           I interviewed Debra Fuentes, one of the coaches and designers of CGI curriculum used at High Tech Elementary Schools. I also interviewed Marissa Funk, a second grade teacher and math coach at High Tech Elementary Explorer. She shared some great ideas about scaffolds for discourse and peer engagement. They shared so many amazing strategies that our team ended up building into the CGI lessons leading up to the research lesson. Some strategies we borrowed were: assigning students “thinking jobs,” (Fuentes, 2019) in which they had to look for similarities and differences between strategies. We also assigned student partnerships to provide safe places to build language. The partners were also able to share thinking at a turn and talk before responding to the comprehension question. The host teacher had started posting student conjectures and recording student language on charts. She also supported language production at discourse with sentence frames. Debra and Marissa were invaluable resources in our thinking about supporting student thinking. The strategies were implemented with some consistency. We were able to see students use the partnered turn and talk. We also were able to see the use of student language for math conjectures. 

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           Now that we had a firm structure, we settled on a content goal about the relationship between subtraction and addition with 3 digit numbers in a Joint start unknown problem (JSU). With a JSU problem, (x+2=5)  there are two things being joined but the story will share the change and the result, but not the start. These problems can be solved with addition or subtraction methods. I felt like our research question had one set of goals, our content derived another goal, and our focus students each had their own personalized goals connected to content and discussion. I was not sure where to go with my research at this point, so we considered the gaps in our research, it became apparent that we needed to deepen our understanding of the content, the concepts needed to meet our goals with invented algorithms and JSU problems, and the expectations we are placing on our focus students. 

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          In searching for a deeper content understanding, I read an article about criteria for assessing invented algorithms (Campbell et all., 1998) . I was excited and anxious by the depth to which we could assess students' invented algorithms. The assessment is in terms of efficiency, validity, and generalizability. There are also the scaffolds for students to assess algorithms and develop agency to do so.  I felt like we could spend a whole other lesson study looking at ways to assess student strategies or even just possible misconceptions. The more I researched, the more I realized how much deeper we could go with our understanding of the content goal. 
Our host teacher had started recording conjectures. I was excited about the idea to have a criteria for the generalizability of a strategy and the opportunity for students’ peers to notice this and develop conjectures inspired by a shared invented algorithm. I hoped students would have the opportunity to make a claim about the relationship between addition and subtraction inspired by their engagement with each others thinking, noticing that some students counted on from the change while others counted down from the result and possible corresponding operations. 

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           My excited anxiety about missing a beat with the depth of our content goals was only slightly exacerbated when I encountered an article about ways to support peer interactions in math, by Dr. Catherine D. Bruce (2007). In one part she celebrates a teacher’s facilitation; stating, “The teacher knew when to intervene and when to let the conversation continue even if it was erroneous.” Would we know when to intervene? Our host teacher had the opportunity to observe a CGI lesson in which an incorrect strategy was shared and we were convinced that sharing one incorrect strategy would support engagement and help the community develop growth mindsets. Personally, I know I am still developing the skill of knowing when to intervene. Writing this after the lesson, I know we could have done some more careful outlining of what exactly we were looking for with an incorrect strategy. We needed a strategy that was on the right track and able to be followed, one that possibly highlights a misconception students have had the opportunity to grapple with before.

 

        The choice of an incorrect strategy might be even more strategically reflective than the sequence or “correct strategies.” The opportunity for students to assess a strategy will help them develop skills they can use with their own work and see connections between patterns in thinking. The opportunity to see the community grow through grappling with a misconception will not only strengthen the community, but the neural pathways that help reinforce the concept. I am excited to deepen this skill in math and science, and know this is dependent on deep content conceptual understandings. I need to spend more time considering misconceptions and finding opportunities for students to carry the cognitive load of proof. 
 
          I read my last resource after the lesson and immediately reflected on something we should consider in the design of future lesson studies, and lessons in general. Reading the addition chapter of Math Matters (Chapin, S. H., & Johnson,  2006). I learned about the tendency of students to resort to less advanced strategies when encountering new challenges. We had goals for each focus student that pushed them to more advanced strategies. Focus students 1 and 4 were able to experiment with more advanced strategies. The host teacher had worked carefully to help build their agency and help them feel safe to try on more challenging thinking. Focus students 1,2, and 4 were able to share their thinking and even engage with peer strategies. The joint start unknown problem type was not completely new to them, but JSU with 3 digit numbers was. We were setting goals that didn’t account for the new problem type. I was thinking about the lesson study structure and how to set goals that students can meet when exploring a new content challenge. We should choose to focus on a new problem type or a risk with a more advanced strategy. Expecting our focus students to meet both goals, while sharing thoughts with the class could have been too much. We are lucky they were comfortable within their community.

           This experience was so humbling and I can not wait to try again.